Motion along a line
In kinematics of linear motion, we measure position from a fixed point O. The displacement s is the signed distance from O: positive on one side, negative on the other. Both velocity v and acceleration a are also signed — a positive value means the positive direction, a negative value the opposite direction.
Reading the functions
Given s, v or a as a function of time t, substitute a value of t to find the quantity at that instant. The initial value is found at t = 0. The particle is momentarily at rest when v = 0, and it is at O when s = 0.
Displacement is not the same as distance. Distance ignores direction and can never decrease, whereas displacement can grow or shrink as the particle moves back and forth about O. Reading the sign of each quantity carefully is the key skill: a negative velocity together with a positive displacement simply means the particle is on the positive side of O but is now heading back towards it.
Key formula /
Initial value: put t = 0. At rest: solve v = 0. At O (starting point): solve s = 0. A negative s means the particle is on the negative side of O.
Worked example
A particle moves so that s = t² − 3t metres. Its initial displacement (t = 0) is 0. At t = 4, s = 16 − 12 = 4 m, so it is 4 m on the positive side of O. It returns to O when t² − 3t = 0, that is t(t − 3) = 0, giving t = 0 or t = 3 s.
Remember
- Sign of s, v, a shows direction.
- Initial value → set t = 0.
- At rest → v = 0; at O → s = 0.